The case \(\delta <1/2\)  and \(\rho =0\)  is not very interesting. If \(k\rightarrow \infty \) , it was remarked in [1]} after Theorem 1 that
\( \lim _{d\rightarrow \infty } \frac{{\mathbb {E}}\,f_k(C_N)}{\binom{N}{k}} =0.\) 
